Abstract
The most important problem in a practical implementation of degressive proportionality is its ambiguity. Therefore, we introduce an order relation on a set of degressively proportional allocations. Its main idea is to define greater allocations such that emerge from other after transferring a certain quantity of goods from smaller to greater entities contending in distribution. Thus, maximal elements in this ordering are indicated as the sought-after solution sanctioning boundary conditions as the only reason of moving away from the fundamental principle of proportionality. In case of several maximal elements the choice of one allocation remains an open issue, but the cardinality of the set from which we make a choice can be reduced significantly. In the best-known example of application of degressive proportionality, which is the apportionment of seats in the European Parliament, the considered set contains a maximal element. Thereby, there exits an allocation that is nearest to the proportional distribution with respect to transfer relation.
1 Introduction
A problem of fair distribution is approached by several disciplines of science in different perspectives. The issue is of interest to philosophers, sociologists, mathematicians, politicians and economists, since there is no unambiguous answer to the question of fairness of a distribution. The answer depends on many factors, with cultural environment in the first place. The concept of fairness in Europe has been formed by Aristotelian principle of proportionality. According to this principle each entity participating in distribution of goods should be allocated a quantity proportional to the value of the entity (participants). The value of entities can be perceived in many ways. In the area of electing political representation it is typically the number of population in an electoral district. When deciding on various issues in joint-stock companies it is capital holdings evidenced by the number of shares owned by shareholders. When determining taxes it is either income or wealth of a given entity.
A rule of proportional allocation however, is not perceived, even in Europe, as the unique solution to the problem of a fair distribution. In particular, its application can be disapproved when the values of participants differ a lot. In such a case the large value of entities can take unfair advantage and be criticized as socially unjust. This problem can be solved by assuming that the entities with greater value reduce their demands for the benefit of entities with smaller value. Thus a classical proportion is distorted for the sake of increasing the portions allocated to those participants whose proportional shares are smallest, at the cost of those with greater shares.
One possible implementation of the mentioned idea is a degressively proportional rule stating that a greater participants cannot be allocated less than a smaller participants, but the quotient of the amount of a good to the participants value cannot increase with the increase of their value.
Definition 1
A positive vector of shares \(S = (s_1, s_2, \ldots , s_n)\) is degressively proportional with respect to positive, nondecreasing sequence of values (demands, claims) \(P = (p_1, p_2, \ldots , p_n)\) if \(s_1 \le s_2 \le \ldots \le s_n\) and \(p_1/s_1 \le p_2/s_2 \le \ldots \le p_n/s_n\).
Let us also introduce: \(s_1=m\), \(s_n = M\), \(\sum _{i=1}^{n}s_i = H\). These three equalities will be henceforth called boundary conditions. In case of degressively proportional allocation they determine the upper and lower bounds for the quantities of goods allocated to participants, and the total quantity of goods.
The most important problem in a practical implementation of degressive proportionality is its ambiguity. Even if we determine the total quantity of an allocated good H, the number of vectors that are degressively proportional with respect to a given sequence of claims can be infinite; an outcome certainly different from the unambiguous solution provided by the rule of proportional allocation. Ambiguity of proportional distribution (allocation) emerges once we distribute indivisible goods, because indivisibility requires \(s_i\) to be expressed as natural numbers, thus rounding is necessary, which can be understood in many ways ([7]). In case of divisible goods a sequence of quota \(Q = (q_1, q_2, \ldots , q_n)\), \(q_i = \frac{p_i H}{\sum _{i=1}^{n}p_j}\), is a unique solution satisfying the rule of proportional distribution.
In this paper, we consider only degressively proportional apportionments in whole numbers with boundary conditions. The problem of ambiguous solutions however is not made simpler. Assuming that \(s_1 = m\), \(s_n = M\) and \(\sum _{i=1}^{n}s_i = H\), we only ensure that the set
of all vectors with natural elements, that are degressively proportional with respect to a vector \((p_1, p_2, \ldots , p_n)\), has a finite number of elements. The number of elements in this set can be arbitrarily large; therefore we have to select one out of many possible allocations.
A natural way to obtain uniqueness is by reference to a fundamental principle of distribution, i.e. to proportional allocation. Considering the boundary conditions one can assume that they are the only factors, which lead to a degressively proportional distribution, and that is why one should seek a solution in the set DP(P, m, M, H) that is nearest to the sequence of quota generated by proportional distribution. In addition, after the boundary conditions are determined, an actual solution can be anticipated. In view of the practice where the boundary conditions are often negotiated among participants, we should consider this interpretation as leading to a compromise.
Allocation of seats in the European Parliament is a practical implementation of division consistent with this idea. The rule itself is stated in the Treaty of Lisbon: “The European Parliament shall be composed of representatives of the Union’s citizens. They shall not exceed seven hundred and fifty in number, plus the President. Representation of citizens shall be degressively proportional, with a minimum threshold of six members per Member State. No Member State shall be allocated more than ninety-six seats” ([9]). The resolution issued by the European Parliament in 2007 specifies among other that “the minimum and maximum numbers set by the Treaty must be fully utilised to ensure that the allocation of seats in the European Parliament reflects as closely as possible the range of populations of the Member States” ([5]). A natural interpretation of these words shows that legislators propose to determine an allocation that is nearest to proportional as a result of degressively proportional approach with boundary conditions \(m=6\), \(M=96\), \(H=751\).
Now we need to formally comprehend nearness of a division with respect to a proportional distribution. The literature includes at least two groups of proposals to solve this problem. First, the application of known methods of proportional allocation exemplified by the Cambridge Compromise ([2, 3]) and maxprop ([1]). Second, the application of elements of operation research that seek an optimum on the set DP(P, m, M, H) with respect to some criterion (see [6, 8]).
In this paper, we propose an approach to finding a solution that differs significantly from the above mentioned. We introduce an order relation on the set DP(P, m, M, H), which is consistent with required proportionality. The main idea of this ordering is to define greater allocations, i.e. such that emerge from other after transferring a certain quantity of goods from smaller to greater entities contending in distribution. Maximal elements in this ordering are indicated as the sought-after solution thus sanctioning boundary conditions as the only reason of moving away from the fundamental principle of proportionality. Maximal allocations in this ordering have the property that entities with greater values minimally reduce their claims subject to constraints imposed by the definition of the set DP(P, m, M, H).
2 Transfer Order
Boundary conditions can be the reason why all elements of the set DP(P, m, M, H) are distant from the proportional distribution. The more distant the values of m and M are from the quota allocated to the smallest (\(\frac{p_1H}{\sum _{i=1}^{n}p_i}\)) and greatest (\(\frac{p_nH}{\sum _{i=1}^{n}p_i}\)) participant, the more remarkable such outcome is.
Example 1
For given \(P=(10,20,50,70)\), \(H=15\), \(m=2\), \(M=5\) a proportional distribution is assured by the vector (1, 2, 5, 7). On the other hand, a set of all degressively proportional distributions has two elements: \(DP\big ((10,20,50,70), 2, 5, 15\big ) = \{(2, 4, 4, 5), (2, 3, 5, 5)\}\).
The three greatest participants in Example 1 are allocated at least the quantity of goods in each degressively proportional distribution as under proportional distribution. This happens, of course, at the expense of the subsequent, fourth, greatest contender. The size of this loss and also the gain of the entity with the smallest value are defined by the boundary conditions \(m = 2\) and \(M = 5\). The two remaining participants can receive either four units of good each or the smaller one – three units, and the greater – five. Therefore a question arises as to which allocation is nearer to the proportional distribution. It is easily seen that in case of the first solution \(s_2 = 2q_2\) holds, i.e. the second participants is allocated twice its proportional share, whereas \(s_3 < q_3\), which implies that this participant does not benefit if we change the allocation rule from proportional to degressively proportional. We also have \(S^* = S + (0,-1,1,0)\), where \(S = (2, 4, 4, 5)\) and \(S^* = (2, 3, 5, 5)\). These relationships can be interpreted accordingly that the allocation \(S^*\) means we take one unit of a distributed good from the second and give it to the third participant. Thus it is a transfer of good from a smaller to a greater one. If we wish to come near a proportional allocation as much as possible, the principle of degressively proportional apportionment requires that we transfer as many units of goods as possible from entities with smaller values to entities with greater values. The following two definitions formalize this concept.
Definition 2
The set of positive transfers is defined as
whereas its elements are called transfers.
Definition 3
A positive transfer relation on a nonempty set DP(P, m, M, H) is called a relation \(\le _{TR_{+}}\) such that
The relationship \(\sum _{i=1}^{n}t_i = 0\) (in Definition 2) ensures that the vectors S and \(S^*\) (in Definition 3) have the equal sums of their elements, hence they represent the allocations of the same quantity of goods. The relationship \(\sum _{i=1}^{k} t_i \le 0\) shows the direction of transfers – from smaller to greater entities. It follows from Definition 1 that undervaluation of entities (under degressively proportional distribution) does not decrease along with the increase of their sizes. Thus, if we want to obtain the allocation, which is nearest to the proportional, the transfer of goods in this direction is right.
Henceforth, instead of a positive transfer relation, we shall briefly refer to a transfer relation. With the data from Example 1 we have \(S^* = S + T\), where \(T = (0, -1, 1, 0) \in TR_{+}\), hence \(S \le _{TR_{+}} S^*\) holds.
A transfer relation can be employed to comparisons of degressively proportional distributions with respect to their nearness to the proportional distribution. It is a result of the following proposition.
Proposition 1
Relation \(\le _{TR_{+}}\) is a partial order relation.
Proof
Reflexivity. For each \(S\in DP(P, m, M, H)\) holds \(S = S + \theta _{n}\), where \(\theta _n = (0,0, \ldots , 0)\), thus we have \(S \le _{TR_{+}} S\).
Antisymmetry. Let \(S \le _{TR_{+}} S^*\) and \(S^{*} \le _{TR_{+}} S\), then \(S^{*} = S + T_1\) and \(S = S^{*} + T_2\) hold for certain \(T_1, T_2 \in TR_{+}\). Hence we have \(S = S + T_1\), or \(T_1 = -T_2\). As the elements of vectors \(T_1\) and \(T_2\) are non-positive, then \(T_1 = T_2 = \theta _{n}\) must hold, thereby \(S = S^{*}\).
Transitivity. Let \(S \le _{TR_{+}} S^{*}\) and \(S^{*} \le _{TR_{+}} S^{**}\), then there exist such \(T_1, T_2 \in TR_{+}\) that \(S^{*} = S + T_1\) and \(S^{**} = S^{*} + T_2\) hold. Therefore, we have \(S^{**} = S + (T_1 + T_2)\). Let \(T_1 = (t_{1,1}, t_{1,2}, \ldots , t_{1,n})\) and \(T_2 = (t_{2,1}, t_{2,2}, \ldots , t_{2,n})\) and we need to show that \(T_1 + T_2 \in TR_{+}\). Since \(\sum _{i=1}^{n} t_{1,i} = 0\), \(\sum _{i=1}^{k} t_{1,i} \le 0\) and \(\sum _{i=1}^{n} t_{2,i} = 0\), \(\sum _{i=1}^{k} t_{2,i} \le 0\) hold for \(k = 1, \ldots , n\), then we have \(\sum _{i=1}^{n} (t_{1,i} + t_{2,i}) = 0\) and \(\sum _{i=1}^{k} (t_{1,i} + t_{2,i}) \le 0\), which means that \(T_1 + T_2 \in TR_{+}\). Hence we obtain \(S \le _{TR_{+}} S^{**}\). \(\square \)
Relation \(\le _{TR_{+}}\) orders the set of degressively proportional apportionments. As a consequence of the previous analysis, if \(S \le _{TR_{+}} S^{*}\) holds, then we acknowledge that \(S^{*}\) is nearer to the proportional allocation than S. On the other hand, relation \(\le _{TR_{+}}\) does not linearly order a set of degressively proportional distributions.
Proposition 2
Transfer relation is not a linear order.
Proof
It is shown by a counterexample. There is given a set DP(P, m, M, H), where \(P = (100, 200, 350, 350, 560, 840, 945)\), \(m=2\), \(M=9\) and \(H=40\). On this basis, we have \(DP(P, m, M, H) = \{A, B, C, D\}\), where \(A=(2, 3, 5, 5, 8, 8, 9)\), \(B=(2, 4, 5, 5, 6, 9, 9)\), \(C=(2, 3, 5, 5, 7, 9, 9)\) and \(D = (2, 4, 5, 5, 7, 8, 9)\). The corresponding Hasse diagram is given in Fig. 1.
Allocations A and B are incomparable under relation \(\le _{TR_{+}}\), since \(A-B = T = (0, -1, 0, 0, 2, -1, 0)\) and \(\sum _{i=1}^{5} = 1 >0\). \(\square \)
As a consequence of Proposition 2, the degressively proportional allocations from the given partially ordered set (poset) \(\big (DP(P, m, M, H), \le _{TR_{+}}\big )\) can be incomparable in some cases. Thus, we are not able to determine which relation from incomparable ones is nearer to the proportional allocation. But we are interested precisely in such allocation, which is the nearest one to the proportional allocation (with respect to a transfer order). If it exists, it is the greatest element of the poset \(\big (DP(P, m, M, H), \le _{TR_{+}}\big )\) that will be called a Transfer Order Allocation (TOA). Proposition 3 shows that the greatest element may not exist.
Proposition 3
There exist P, m, M, H for which a poset (DP(P, m, M, H), \( \le _{TR_{+}})\) does not contain the greatest element.
Proof
We prove the proposition by a counterexample. Given the set DP(P, m, M, H), where \(P=(100, 200, 466, 466, 931, 1165)\), \(m=2\), \(M=10\), \(H=35\), we have \(DP(P, m, M, H) = \{A, B\}\), where \(A = (2, 3, 6, 6, 8, 10)\) and \(B=(2, 4, 5, 5, 9, 10)\). Note that \(A-B = T = (0, -1, 1, 1, -1, 0)\), therefore, we have \(\sum _{i=1}^{2} t_i = -1\) and \(\sum _{i=1}^{4} t_i = 1 > 0\), which means that T is not a transfer (see Definition 2). On the other hand, for \(B-A = T' = (0, 1, -1, -1, 1, 0)\), we have \(\sum _{i=1}^{2} t'_i = 1\), which is not a transfer. Therefore, allocations A and B are incomparable under the relation \(\le _{TR_{+}}\), thereby the existence of two maximal elements results in the lack of the greatest element. A related Hasse diagram is given in Fig. 2. \(\square \)
When the greatest element does not exist, the transfer order does not unambiguously point to the allocation from the set DP(P, m, M, H) as the nearest one to the proportional distribution. However, if the greatest element exists, it is easily obtained by Proposition 4. Before expressing this proposition, we shall prove the following lemma.
Lemma 1
The greatest element in the poset \((DP(P, m, M, H), \le _{AL})\), where \(\le _{AL}\) is an antilexicographic order, is the maximal element in the poset \((DP(P, m, M, H), \le _{TR_{+}})\).
Proof
Let us assume the opposite that S is not a maximal element in the set DP(P, m, M, H) with transfer order. Then there exists such element \(S^{*}\) that \(S \le _{TR_{+}} S^{*}\) holds. This means that there exists such transfer T that \(S^{*} = S + T\) holds. It follows then from Definition 2 that \(S^{*}\) is also greater than S under antilexicographic order, hence S is not the greatest element in the set DP(P, m, M, H) under antilexicographic order. \(\square \)
Proposition 4
If there exists a greatest element in the set DP(P, m, M, H) with transfer order, then it is also the greatest element in the same set with antilexicographic order.
Proof
It results directly from Lemma 1. \(\square \)
3 Case Study
The acts of law, which introduce and regulate the distribution of seats in the European Parliament, determine the set DP(P, 6, 96, 751), where \(P=(p_1, p_2, \ldots , p_n)\) is the vector of populations of the Member States. There are 751 seats, which are distributed among 28 states in a degressively manner, with the least populated country receiving 6 seats and the most populated – 96 seats. Obviously, as a result of the assumed boundary conditions countries with less population are allocated more seats than under proportional allocation, and countries with more population are given fewer seats than under proportional scheme. For example, proportional allocation would assign no more than one seat to small countries such as Malta, Luxembourg or Cyprus, while Germany would be assigned at least 120 seats. However the smallest country and the greatest country are allocated precisely 6 and 96 seats respectively, the numbers of seats for remaining states are not determined uniquely. What is more, with data on populations in 2012 (employed to determine the composition of the European Parliament in the current term of 2014–2019), the set DP(P, 6, 96, 751) has cardinality more than 5 million (see [6]). Undoubtedly therefore, a rule must be indicated that will allow the choice of a specific allocation.
It turns out that the set DP(P, 6, 96, 751) contains the greatest element with respect to transfer order, i.e. the transfer order proposed in the previous section allows finding the allocation that is nearest to the proportional one. This allocation denoted by TOA and presented in column 9 of Table 1 reduces the undervaluation, imposed by the definition of the set DP(P, m, M, H), of most populated countries for the benefit of less populated countries compared to proportional allocation. Thus in contrast to other presented allocations, TOA gives more populated countries more seats. Therefore it ensures both the satisfaction of postulates of parliamentarians requiring the best possible representation of differences in populations of the Member States as well as the compliance with current regulations.
Table 1 presents several selected allocations. Columns 4 and 5 contain numbers of seats allocated to countries for the current term of the European Parliament in 2014–2019. Bold entries in column 5 indicate violations of the degressive proportionality principle. Nevertheless, the parliamentarians agreed to the incompliance of this solution with currently binding law (see [4]).
Column 6 shows the distribution in compliance with the Cambridge Compromise ([2]), which is determined by the rule \(s_i = \max \{5+\lceil p_i/a\rceil \}\), where a constant a is set to ensure that the total of all distributed seats equals 751 (for example \(a\in [839.94, 844.30]\) for data from 2012). This distribution, similarly as current allocation, violates the principle of degressive proportionality (bold entries in column 7). This incompliance results from the design of the procedure that can return an allocation, which is not an element of the set DP(P, 6, 96, 751) (for more details see [2]). The allocations presented in columns 8 and 9 are free of that flaw.
Column 8 contains the allocation generated by LaRSA algorithm ([6]), which consists in searching the set DP(P, 6, 96, 751) and selecting the allocation that minimizes the sum of squared distances form the proportional allocation, i.e. \(\sum _{i=1}^{n}\big ( p_iH/\sum _{j=1}^{n} p_j\big ) \rightarrow \min \) .Column 9 contains the TOA, the greatest element of the set DP(P, 6, 96, 751) with transfer order. By Proposition 4, this allocation is also the greatest element in the set DP(P, 6, 96, 751) with antilexicographic order. As we can see, the LaRSA and TOA allocations are identical. However that is not necessarily the case. There are some examples showing that the allocations generated by those methods are different. In case of the European Parliament though, the equality of those two allocations additionally supports the idea of coming closer to the proportional allocation by means of transfer of seats. Obviously both approaches ensure that the degressively proportional allocation is obtained, because they are selected from the set DP(P, 6, 96, 751).
4 Conclusions
A transfer relation orders a given set of degressively proportional allocations. Except for cases when it is not a linear order, the relation makes it possible to specify which allocation is nearer to the proportional distribution, and particularly to determine the maximal elements, i.e. such that no other distribution is better. If there is a greatest element in the given set DP(P, m, M, H) with transfer order, we assume that this is the optimal allocation. In addition, there is an alternative method to find it – by Proposition 4 it is the greatest element in the set DP(P, m, M, H) with antilexicographic order. In case of several maximal elements the choice of one distribution remains an open issue, but the cardinality of the set from which we make a choice can be reduced significantly. In the best-known example of application of degressive proportionality, i.e. the apportionment of seats in the European Parliament, the set DP(P, m, M, H) contains a maximal element. In other words, there exits an allocation that is nearest to the proportional distribution with respect to transfer relation.
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Acknowledgement
The results presented in this paper have been supported by the Polish National Science Centre under grant no. DEC-2013/09/B/HS4/02702.
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Cegiełka, K., Dniestrzański, P., Łyko, J., Maciuk, A., Rudek, R. (2017). On Ordering a Set of Degressively Proportional Apportionments. In: Mercik, J. (eds) Transactions on Computational Collective Intelligence XXVII. Lecture Notes in Computer Science(), vol 10480. Springer, Cham. https://doi.org/10.1007/978-3-319-70647-4_4
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